from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2019, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,13]))
pari: [g,chi] = znchar(Mod(8,2019))
Basic properties
| Modulus: | \(2019\) | |
| Conductor: | \(2019\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2019.s
\(\chi_{2019}(8,\cdot)\) \(\chi_{2019}(161,\cdot)\) \(\chi_{2019}(209,\cdot)\) \(\chi_{2019}(464,\cdot)\) \(\chi_{2019}(512,\cdot)\) \(\chi_{2019}(665,\cdot)\) \(\chi_{2019}(1262,\cdot)\) \(\chi_{2019}(1430,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{16})\) |
| Fixed field: | Number field defined by a degree 16 polynomial |
Values on generators
\((674,1351)\) → \((-1,e\left(\frac{13}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2019 }(8, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(-i\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(-1\) | \(i\) | \(-1\) |
sage: chi.jacobi_sum(n)